A counterexample to Liu's conjecture on Fano indices in dimension four: paper, Lean 4 formalization and verification code

We prove that Liu's conjecture on Fano indices (H. Liu, arXiv:2608.03434, Conjecture 4.2) fails in dimension four: 1389 is the index of a terminal Calabi-Yau fourfold and of a four-dimensional log canonical singularity that is not klt, both built from a symmetry of order 1389 of the hypersurface x0^77 x1 + x1^67 x2 + x2^7 x0 + x3^3 + x4^2 = 0 in P(1,1,11,26,39), but it is not the Fano index of any canonical weighted projective space of dimension four.

preprintLiuFanoIndexConjecture/ holds the paper as a PDF; paper/ holds the LaTeX source (amsart) with three schematic figures; dist/ holds the PDF, an anonymized PDF for double-anonymous review, the LaTeX source zip and the arXiv tarball; MasterLiuFanoIndexConjecture_masterzipfinal2.zip holds every file of the repository. lean/ is a Lean 4 and Mathlib project proving 1389 notin I_{4,wps}, with Kasprzyk's criterion as the definition of canonical, through a certificate of 4,022 units checked by decide +kernel, and the algebraic lemmas on the hypersurface; its theorems depend only on the axioms propext, Classical.choice and Quot.sound. code/ contains exhaustive searches over all 1,301,769,806 partitions of 1389 into five parts in C, C++, Julia and Python, and scripts/run_all.sh regenerates the certificate, builds the Lean project, checks the axioms and runs all searches. The geometric steps (Kasprzyk's criterion, weighted hypersurfaces, the equivariant terminalization, the singularities) are proved by hand in the paper.